Indirect Instantaneous Car-Fuel Consumption Measurements
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Postprint: to appear in IEEE Trans Instrumentation and Measurement
IEEE TRANSACTIONS , VOL. X, NO. X, XXXX 200X 1
Indirect Instantaneous Car-Fuel Consumption
Measurements
Isaac Skog Member, IEEE and Peter Händel, Senior Member, IEEE
Abstract—A method to estimate the instantaneous fuel con- Data logger
sumption of a personal car, using speed and height data recorded
by a global positioning system (GPS) receiver and vehicle
parameters accessible via national vehicle registers and databases
on the world wide web, is proposed. The method is based upon a GPS receiver OBD device
physical model describing the relationship between the dynamics
of the car, the engine speed, and the energy consumption of
the system. An evaluation of the proposed method is done by
comparing the estimated instantaneous fuel consumption with
that measured by the car’s on-board diagnostics (OBD) data
bus. The results of three tests with different cars driven in mixed
highway and urban conditions, indicate that the instantaneous
fuel consumption may be estimated with a root mean square
error of about 0.3 [g/s]; in terms of a normalized mean square
error, that corresponded to slightly less than 10 percent. One Fig. 1: The measurement equipment used to evaluate the proposed fuel
application of the proposed method is in the development of consumption estimation method. The speed and height data recorded by the
smartphone applications that educate drivers to drive more fuel GPS-receiver is used to drive the fuel consumption model, whereas the engine
efficiently. speed, mass air flow rate, and fuel to air ratio recorded from the OBD bus
are used as reference data.
I. I NTRODUCTION
a fuel consumption map, 1 together with sensors to observe
The number of cars in the world is predicted to double by the speed of the car, the engine speed, gear position, etc.
the year 2035, according to World Energy Outlook 2011 [1]. However, if such a system should have the potential to reach a
The emissions of those predicted 1.7 billion cars will induce large group of drivers, it must be cheap and easy to use. This
large environmental challenges. Consequently, there are in- means that the system should be able to run on already existing
tense research and development activities in the areas of non computational platforms, e.g., a smartphone, and be based
fossil energy sources and low emission vehicles. These are upon indirect measurements of the fuel consumption so that
also the areas where in the long run, the largest contributions the trouble of connecting the system to the vehicle’s onboard
to the overall reduction in emissions and fuel consumption of computer is avoided. Another important application is found
the car fleet are forecasted. However, in the short run, while in the developing nations where the fuel cost is one of the
waiting for tomorrow’s technology, traffic management and a major costs of running a vehicle fleet. Many vehicles, because
change in driving behaviors may significantly contribute to of their age or the down-stripping of the electronics by the
lowering fuel consumption and emissions [2]. manufacturers, do not provide the relevant data for calculating
One way to teach a driver about energy-efficient driving the actual fuel consumption via the on board diagnostic outlet.
is through eco-driving courses, something which today is In this case, a smartphone can play the role of a measurement
mandatory in order to obtain a driving licence in several device to track the fuel consumption to reduce not only the
European countries. However, the effect of an eco-driving direct fuel consumption, but also to detect fraud or tampered
course on a driver’s behavior diminishes as time goes by, pumps at filling stations.
and regular monitoring and feedback is required to maintain Work on indirect measurements appear quite frequently in
the effects of the eco-driving training [3], [4]. Thus, a system the instrumentation and measurement literature, since from a
that can give the driver immediate and constructive feedback cost, practical, or safety perspective, it is often undesirable
about how their actions affect the fuel consumption is needed, to connect and interact with the system undergoing testing.
to generate a long-term change in the behavior of a driver. To give a few examples: a method to indirectly estimate the
In [2], such a feedback system is presented and their test timing and duration of the manual gear shift in a personal
results show an up to 23% reduction in fuel consumption. car using accelerometer measurements and a piecewise linear
As a key component for generating the feedback, they use model of the car’s acceleration were considered in [5]; a
method to measure car performance in terms of quantities
Manuscript received
I. Skog and P. Händel, are with the ACCESS Linnaeus Center, Dept. of such as elapsed time and speed during drag racing activities,
Signal Processing, KTH Royal Institute of Technology, Stockholm, Sweden.
(e-mail: isaac.skog@ee.kth.se; ph@kth.se). 1 A fuel consumption map is a three dimensional plot of the specific fuel
Copyright (c) 2010 IEEE. consumption versus the engine rotation speed and brake torque.Postprint: to appear in IEEE Trans Instrumentation and Measurement
IEEE TRANSACTIONS , VOL. X, NO. X, XXXX 200X 2
TABLE I: Physical constants and car model independent parameters used in
using accelerometer measurement and a model of the chassis the fuel consumption estimator.
squat during acceleration were considered in [6]; a method to
indirectly measure railroad-curvature by fusing GPS-receiver, Physical constants
gyroscope, and speed measurements were considered in [7]; Parameter Unit Value Description
pedestrian activity classification using measurements from g m/s2 9.806 Standard gravity
chest-mounted inertial measurement units and a model for ρa kg/m3 1.225 Mass density of air
a set of pre-defined gait activities were considered in [8]. Fixed model parameter
The idea behind all the mentioned examples is to, via model
Parameter Unit Value Description
based signal processing relate a measured quantity to another
QL J/kg 44.4 · 106 Lower heating value of gaso-
quantity that cannot be directly measured, i.e., to perform line
indirect measurements. ηt − 0.96 Transmission efficiency
One approach to design a system to instantaneously estimate ηc − 0.98 Combustion efficiency
ηig − 0.31 Gross indicated thermal effi-
the fuel consumption of a car indirectly, is thus to, from phys- ciency
ical laws, deduce a model that relates the motion dynamics of Cr − 0.013 Roll resistance
a car to its fuel consumption. In [9] and [10], such physical c1 N/m 9.7 · 104 Total friction mean effective
pressure coefficient
models are deduced and used for emission modeling in micro- (N/m)
c2 (rev/s)
900 Total friction mean effective
scale traffic simulations. However, these models assume that pressure coefficient
several car model specific parameters are known, and that not (N/m)
c3 (rev/s)2
18 Total friction mean effective
only the motion dynamics of the car are observed, but also the pressure coefficient
engine speed. These models can thus, not without modification Pa W 1500 Power consumption of the ac-
cessors in the car
and prior knowledge about the technical parameters of the car, ridle rev/s 13 Idle engine speed
be used to estimate the fuel consumption of the car, purely
from its motion dynamics.
Therefore, in this paper we investigate the possibility of
extending the physical model in [9], with a model for the A. Power flow model
engine speed, and to extract the necessary model parameters Let Pt [W ] denote the instantaneous tractive power needed
from the Swedish Traffic Registry 2 , and thereby estimate the at the wheels for the car to obey the motion change com-
fuel consumption of a car solely from GPS-receiver recordings manded by the driver. Further, let P tf [W ] denote the power
of its speed and height dynamics. The accuracy of the proposed required to overcome the total engine friction, and P a [W ] the
fuel consumption estimation method is evaluated by compar- power needed to drive accessories such as the air-conditioner,
ing the estimated fuel consumption with the fuel consumption etc. Next, assuming that the driver never has the clutch
readout from the car’s on-board diagnostics (OBD) port, see engaged when ever P t < 0, then all power generated from
Fig. 1. The results from three test drives in mixed highway the wheels will be dissipated through the brakes 3 . We can
and urban conditions, with three different car models, indicate then model the required instantaneous gross indicated power,
that the proposed method can estimate the fuel consumption Pig [W ] as
with a root mean square error of the order of 0.2-0.4 [g/s];
in terms of normalized mean square error, that corresponds to 1
Pig = max(Pt , 0) + Ptf + Pa . (1)
slightly less than 10 percent. ηt
Here, ηt [−] denotes the efficiency of the transmission (in-
cluding the final drive). The efficiency of the transmission
II. P OWER SOURCES AND LOADS
depends on several parameters, such as engine speed, torque,
To develop a model for the instantaneous fuel consumption gear-ratio, temperature, etc., and thus varies between different
of the vehicle, we need models for the various power sources car models and with the driving conditions [11], [12]. In
and loads in the vehicle, as well as a model for the power [12], the calculated average efficiency for a five-speed manual
flow between them. Hence, we will start by introducing a transmission, varied from 92%-97% depending upon the gear.
simple power flow model. Thereafter, we will present models In our model, we will use a value on the upper end of the
for the different power sources and loads. We will develop scale, i.e., ηt = 0.96. The values of all physical constants
the models to resemble the properties of cars with manual and parameters used in the fuel consumption estimator are
transmissions and electronically controlled injection gasoline summarized in Table I.
engines. However, the models will be sufficiently generic to be
easily adapted to cars with automatic transmissions and those B. Gross indicated power versus fuel mass flow rate
that run on diesel, E85, etc.
We can relate the gross indicated power in (1) to the fuel
2 The Swedish Traffic Registry is a publicly accessible register that contains
mass flow rate (fuel consumption) y [g/s] via [13]
information about the current keeper of the vehicle and vehicle data such as
brand, model, vehicle class, engine capacity, wheel diameters, etc. Similar 3 Note that, even though power is a positive quantity, the idea of negative
registers also exist in many other countries, e.g., in Denmark, Norway, and tractive power is used throughout the paper to mathematically model the fact
Finland. that when the car is decelerating, the wheels may be used as a power source.Postprint: to appear in IEEE Trans Instrumentation and Measurement
IEEE TRANSACTIONS , VOL. X, NO. X, XXXX 200X 3
fa
x
p
fr + fa - Air drag force
Pig
y= . (2) fr - Roll resistance force
QL ηc ηig ft - Tractive force
fg ft fg - Gravity force
Here, QL , ηc , and ηig denote the lower heating value of ft
the fuel, the combustion efficiency, and the gross indicated p
z xn (north)
efficiency, respectively. The lower heating value of the fuel
varies with the fuel composition. The combustion efficiency
z n (down)
and the gross indicated efficiency varies with parameters
Fig. 2: Illustration of the major forces acting upon the car (red arrows), the
such as the equivalency ratio and sparking time [13], [14]; vehicle (platform) coordinate system (black arrows), and the local geodetic
parameters that cannot be deduced from the data output by coordinate systems (black arrows).
the GPS-receiver. We will therefore, use the following constant
values for the parameters, Q L = 44.4 · 103 [J/g], ηc = 0.98,
and ηig = 0.31 [15, p.155]. where Cr [−] is the roll resistance coefficient of the tires.
Further, θ and φ are the slope of the road in the longitude
and lateral directions, respectively. In general, the slope of the
C. Motion dynamics versus tractive power
road is less than 10 percent, and we will approximate the roll
Next, we will derive a simple model for the tractive power resistance force as
Pt , needed for the car to obey the driver’s commanded motion
change. The model is similar to the models presented in [9], [frp ]x Cr m g. (7)
[10], and [16], and is deduced from Newton’s second law of
motion and the major forces acting upon the car. The major The air-drag force can be modeled as
forces acting upon the car are the tractive force, the air drag ρa
force, the roll-resistance force, and the gravity force. We have A Ca (vxp )2 ,
[fra ]x = (8)
2
illustrated these forces and their directions in Fig. 2. The
where Ca [−] is the air resistance coefficient, ρ a [kg/m3 ] the
change in velocity, v [m/s], of a vehicle with the mass, m
mass density of the air, and A [m 2 ] the cross area section
[kg], as a function of the sum of these forces is
of the vehicle. Inserting (7) and (8) into (5), we get that the
dvp tractive power needed to induce the change in velocity can be
m = ftp − fap − frp − fgp . (3)
dt described by
Here, ft [N ], fa [N ], fr [N ], and fg [N ] denote the tractive
ρa
force, the air drag force, the roll-resistance force, and the Pt = m apx vxp + Cr m g vxp + A Ca (vxp )3 − m g vzn , (9)
gravity force, respectively. The superscript p denotes that 2
dv p
quantity is expressed with respect to the vehicle’s (platform) where apx dtx and vzn [vn ]z . Thus, if the vehicle specific
coordinate system. The tractive power, P t [W ], needed to parameters, mass m, air resistance coefficient C a , and cross
induce the velocity change is area section A, and the environmental and tire dependent roll
resistance coefficient Cr , are known, then the tractive power
Pt = ftp · vp Pt can be estimated from the motion dynamics a px , vxp , and vzn .
dvp An illustration of the magnitude of the different components
= (m + fap + frp + fgp ) · vp (4) in (9) are shown in Fig. 3.
dt
dvp Out of these parameters, only the vehicle’s curb weight 4
= (m + fap + frp ) · vp + fgn · vn . information is available in the Swedish Traffic Registry. How-
dt
ever, the product C a A varies little between cars in the same
Here, a · b denotes the dot-product between the vectors class (segment). Therefore, given the class the car belongs
a and b, and the superscript n denotes that a quantity is to (information that can be found in the Swedish vehicle
expressed with respect to the local geodetic coordinate system register), we approximate C a A with the average value given
(north, east, and down). Next, note that f gn = [ 0 0 − m g ], in Table II. The rolling resistance of a car’s tires depends
where g [m/s2 ] is the magnitude of the local gravity vector. on several factors, their rotation speed, their temperature, and
Further, assume that the car experiences no sideslip, i.e., the texture of the road [17]. For modern car tires, the rolling
vp = [ vxp 0 0 ], where vxp [m/s] is the along-track speed of resistance ranges, approximately from 0.008 on a smooth-
the car. Then, (4) simplifies to textured surface, to 0.018 on rough-textured surface [18]. In
our model, we will therefore assume that C r = 0.013.
dvp
Pt = m [ ]x + [fa ]x + [fr ]x vxp − m g[vn ]z .
p p
(5)
dt D. Engine friction versus engine rotation speed
Here, [u]i , i = {x, y, z}, denotes the i:th element of the vector The total friction mean effective power P tf [W ], needed to
u. The roll resistance force can be modeled as do the pumping of the fuel in the cylinders, to run the engine
4 Total weight of a vehicle with standard equipment, all necessary operating
[frp ]x = Cr m g cos(θ) cos(φ), (6) consumables, a full tank of fuel, and a 75 kg heavy driver.Postprint: to appear in IEEE Trans Instrumentation and Measurement
IEEE TRANSACTIONS , VOL. X, NO. X, XXXX 200X 4
TABLE II: Engine volume, peak engine power, gear-ratio, and air resistance specifications for seven gasoline engine cars in the C segment.
Model Vd [cm2 ] Pbpeak (r ∗ ) [kW ] @ (r∗ [rev/s]) GL GF GF GL L Ca A [m2 ]
1.6 Liters
Audi A3 1.6 FSI (2005) 1598 85 (100) 0.71:1 4.53:1 3.22:1 6 0.70
Kia Cerato Hatch 1.6 (2011) 1591 91 (105) 0.73:1 4.27:1 3.11:1 6 –
Citroën C4 1.6i 16v (2004) 1587 80 (97) – – – 5 0.68
Ford Focus 1.6i (2007) 1596 85 (100) 0.88:1 4.06:1 3.57:1 5 0.73
Seat Toledo 1.6 (2004) 1596 75 (93) 0.85:1 4.53:1 3.85:1 5 –
Skoda Octavia 1.6 FSI (2004) 1598 84 (100) 0.78:1 4.53:1 3.53:1 5 –
VW Golf 1.6 FSI (2003) 1598 84 (100) 0.71:1 4.53:1 3.22:1 6 0.71
Average values – 83 (99) 0.78:1 4.41:1 3.42:1 - 0.71
2.0 Liters
Audi A3 2.0 FSI (2003) 1984 110 (100) 0.82:1 3.65:1 2.99:1 6 0.66
Kia Cerato Hatch 2.0 (2011) 1998 115 (103) 0.76:1 4.19:1 3.18:1 6 –
Citroën C4 2.0i 16v (2007) 1997 103 (100) – – – 5 0.70
Ford Focus 2.0 (2011) 1999 119 (108) 0.81:1 3.82:1 3.09:1 5 –
Seat Toledo 2.0 FSI (2004) 1984 110 (100) 0.87:1 3.94:1 3.43:1 6 –
Skoda Octavia 2.0 FSI (2004) 1984 110 (100) 0.82:1 3.65:1 2.99:1 6 –
VW Golf 2.0 FSI (2003) 1984 110 (100) 0.82:1 3.65:1 2.99:1 6 0.71
Average values – 111 (102) 0.82:1 3.82:1 3.11:1 - 0.69
vzn [m/s]
ther, nR denotes the ratio between the number of strokes of
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
30 the pistons to the number of rotations of the crankshaft, i.e.,
Change in motion energy m a px vxp ; apx = 1
Rolling resistance Cr m g vxp nr = 1 for a two stroke engine and n R = 2 for a four stroke
Air drag ρ2a A Ca (vxp )3 engine. A simple model for the total motored friction mean
Engine friction P tf (r)
25
Change in potential energy m g v zn effective pressure k tmf (r) [N/m2 ] is given by [14], [19]
ktmf (r) = c1 + c2 r + c3 r2 . (11)
20
In [14, pp.722-723], the coefficients that best fit (11) with
Power [kW ]
the measured total motorized friction mean effective pressure
15 for several four stroke engines with the displacement volumes
ranging from 845-2000 [cm 3 ] were identified as c1 = 9.7·104,
c2 = 900, and c3 = 18. In our fuel consumption estimator, we
10 will assume that ktf (r) ≡ ktmf (r).
E. Accessories power consumption
5
Generally, the most power demanding accessory in a mod-
ern car is the air-conditioner, which, for a sedan at peak power,
can induce a load of 5-6 [kW ] on the engine [20]. The power
0
10 20 30 40 50 60 70 80 90 100 consumed by the air-conditioner to generate a climate that
vxp [km/h] or r [rev/s] satisfies the driver, depends on a number of factors such as
Fig. 3: Illustration of the magnitude of the different terms in the fuel the solar radiation, ambient air temperature, air humidity, and
consumption model for a vehicle with a 1.6 [dm3 ] four-stroke engine, mass number of people in the car. As these factors are obviously not
m = 1000 [kg], and an air drag factor Ca A = 0.7 [m2 ]. Remaining model deducible from the data from the GPS-receiver, we will assume
parameters are given in Table I.
a constant value, P ac = 1.5 [kW ], for the power consumption
of the accessories in the car.
accessories necessary for the functionality of the engine, and III. S YNTHESIZING THE ENGINE ROTATION SPEED
to overcome the rubbing friction is given by
In the previous section models for the power sources and
Vd r load in vehicle were presented. The magnitude of one of
Ptf = ktf (r) . (10) them, the power needed to overcome the engine friction,
nR
depends on engine speed. Since the engine speed is not directly
Here, ktf (r) [N/m2 ], Vd [m3 /rev], and r [rev/s] denote the measurable using a GPS-receiver, we will in this section
total friction mean effective pressure, the engine displacement, present a way to synthesize the engine speed using knowledge
and the engine (crankshaft) rotation speed, respectively. Fur- about the speed of the vehicle, the vehicles engine size, etc.Postprint: to appear in IEEE Trans Instrumentation and Measurement
IEEE TRANSACTIONS , VOL. X, NO. X, XXXX 200X 5
Maximum tractive power versus engine rotation speed
The speed of the car is related to the rotation speed of the 150
engine, r [rev/s], according to Volvo (Measured)
Volvo (Estimated)
r Audi (Measured)
vxp = φw , = 1, . . . , L, (12) Audi (Estimated)
GF G Peugeot (Measured)
Peugeot (Estimated)
where φw [m/rev], GF [−], and G [−] denote the cir- 100 Ford (Measured)
cumference of the wheels, the final-drive gear-ratio, and the
Ptmax (r) [kW ]
Ford (Estimated)
transmission gear-ratio at gear .
Let, L be the number of the highest gear (top gear), then a
lower bound on the engine speed is given by
50
vp
r ≥ rspeed , rspeed = GF GL x . (13)
φw
Another lower bound on the engine speed may be found from
the tractive (wheel) power P tmax (r) curve, that specifies the
0
maximum power that can be delivered at the wheels at a given 10 20 30 40 50 60 70 80 90 100 110
r [revs/s]
engine speed. Since P tmax (r) must be greater than or equal
to the needed tractive power P t , we have that Fig. 4: Measured and estimated tractive (wheel) power curves for a Ford
Fiesta 1.25 (Pepeak = 55 [kW ]), a Peugeot 307 1.6 (Pepeak = 81 [kW ]), an
Audi A3 1.8T (Pepeak = 110 [kW ]), and a Volvo V70 2.4T (Pepeak = 147
r ≥ rpower , rpower = argmin(Ptmax (r) ≥ Pt ). (14) [kW ]).
r
By combining (13) and (14), we get that
agreement between the measured and estimated curves shows
that the model in (16) captures the main characteristics of the
r ≥ rmin , rmin = max(rspeed , rpower , ridle ), (15)
brake power curve of an engine, and that we may use it for
where ridle is the engine rotation speed at idle. evaluating the lower bound for the engine’s rotational speed.
The Swedish vehicle register only holds information re- For more refined models for the power curve of an engine,
garding the circumference of the wheels and the engine’s see e.g., [22].
peak power P epeak [W ], and (15) therefore, cannot directly
be used to lower bound the rotational velocity of the engine. IV. C ALCULATING THE MODEL INPUTS FROM GPS- DATA
However, as seen from the vehicle data in Table II, the product
GL GF varies little between cars in the same class. Further, The model for the fuel consumption (fuel mass flow)
the rotational velocity of the engine when at idle, is for most developed in the previous sections is driven by three inputs,
gasoline car engines between 10-16 [revs/s]. Therefore, in our the along track speed v xp , the along track acceleration a px ,
model we will use the average value in Table II for the product and the vertical velocity v zn . It may seem straightforward to
GL GF , and let ridle = 13 [rev/s]. To find rpower , we will calculate these quantities by simply differentiating the speed
assume that the power curve of the engine is approximately and height measurements of the GPS-receivers. However, auto-
described by the following piecewise linear function motive applications provide a challenging environment for the
reception of GPS signals due to factors like antenna placement,
⎧ potential interferences from embedded electronics, and signal
⎪ 9 peak
⎨ 800 Pe r, r ≤ 80 multipath and obstruction, which may cause large errors in
1
Pe (r) = 200 Pe 1
max peak
r + 2 Pe , 80 < r < 100 . (16)
peak
the measurements [23]. When differentiating the speed and
⎪
⎩ peak
Pe r ≥ 100 height measurements, the high frequency error components in
the measurements are amplified, and the calculated along track
That is, between 0-80 [rev/s], the maximum power the acceleration and vertical velocity may become very noisy or
engine can deliver increases linearly with the engine’s rotation distorted by outliers [24].
velocity, until 90% of P epeak . Thereafter, between 80-100 One method that has successfully been used to address these
[rev/s] the maximum power the engine can deliver increases problems, see e.g. [25], is the polynomial regression method,
at a much lower rate with the engine’s rotation velocity, until in which it is assumed that the speed and the height of the
Pepeak . Above 100 [rev/s], the maximum power the engine vehicle locally are described by polynomial models. That is,
can deliver is constantly P epeak . In Fig 4, the measured tractive the k:th speed measurement provided by the GPS-receiver is
(wheel) power curves for a Ford Fiesta 1.25 (P epeak = 55 modeled as
[kW ]), a Peugeot 307 1.6 (P epeak = 81 [kW ]), an Audi
A3 1.8T (Pepeak = 110 [kW ]), and a Volvo V70 2.4T p
ṽx,k = vxp (tk ) + wk , (17)
(Pepeak = 147 [kW ]) are shown; the measured tractive power
curves are based upon the data found in [21]. Also shown are where
the tractive (wheel) power curves estimated using the engine
power model in (16), i.e., P̂tmax (r) = ηt Pemax (r). The good vxp (t) = α0 t2 + α1 t + α2 , t ∈ [tk − T /2, tk + T /2]. (18)Postprint: to appear in IEEE Trans Instrumentation and Measurement
IEEE TRANSACTIONS , VOL. X, NO. X, XXXX 200X 6
TABLE III: Vehicle parameters obtained from the Swedish Traffic Register
Here, tk [s] denotes the sample time of the k:th measurement. for the three vehicles used in the evaluation of the proposed fuel estimation
Further, wk [m/s] denotes the noise in the measurements and method.
T [s] is the time period for which the polynomial model may
be considered valid. The N measurements taken in the interval Parameter Unit Peugeot 206 Audi A3 Volvo XC70
[tk − T /2, tk + T /2] can then be modeled as m kg 1038 1360 1730
Vd dm3 /rev 1.6 1.8 2.4
φw m/rev 1.870 2.155 1.993
zk = Hk θ + wk , (19)
Pepeak kW 80 110 147
Segment (EU) – B C DE
where
p p p
zk = ṽx,k− N
2
. . . ṽx,k . . . ṽx,k+ N
2
, (20)
⎡ ⎤ methods to track the motion dynamics of cars using a plurality
1 (tk− N − tk ) (tk− N − tk )2
⎢ 2 2
⎥ of sensors can be found.
⎢ .. .. .. ⎥
⎢ . . . ⎥
⎢ ⎥
Hk = ⎢ 1 0 0 ⎥, (21) V. M ODEL EVALUATION METHODOLOGY AND RESULTS
⎢ ⎥
⎢ .. .. .. ⎥ To evaluate the proposed fuel consumption estimation
⎣ . . . ⎦
method, we tested it on three vehicles: a Peugeot 207, an Audi
1 (tk+ N − tk ) (tk+ N − tk )2
2 2 A3, and a Volvo XC70. In Table III, the vehicle parameters
available in the Swedish Traffic Register for these three vehi-
wk = wk− N
2
. . . wk . . . wk+ N
2
, (22) cles are shown. We drove these three vehicles along different
and trajectories of length 27-32 km; trajectories that consisted of
a mixture of highway and urban roads. During these drives,
we used the GPS and OBD data logger shown in Fig. 1 to
θ= α0 α1 α2 . (23)
record the engines’ rotation speed, air flow rate, and air-to-
Here, (·) is used to denote the transpose operator. The fuel ratio, as well as the speed and height measurements from
coefficients of the polynomial model (18) can be estimated the GPS-receiver. All the data was recorded at a rate of 5 Hz.
using the weight least squares method. That is, The data was then processed in accordance with the system
illustrated in Fig. 5. That is, given the registration plate
θk = (HTk Q−1
k Hk )
−1 T −1
Hk Qk
zk , (24) numbers of the cars, we downloaded the vehicle parameters
available in the Swedish vehicle register, which together with
where Qk [(m/s)2 ] denotes the the covariance p
of the measure- the average parameter in Table I were used as parameters in
dvx (t)
ment error vector w k . Now, since apx (t) = dt = 2 α0 t+α1 , our fuel consumption model. From the recorded GPS speed
then the acceleration of the vehicle at time t k according to and height measurements, we then, using the polynomial
the polynomial model is given by â px,k = [θk ]2 . Further, by regression method described in Section IV, estimated the along
the same reasoning, the speed v̂ x,k p
= [θk ]3 . The vertical track acceleration â px,k , along track speed v̂ x,k
p
, and vertical
n n
velocity vz may be estimated in a similar way by assuming speed v̂z,k . From these estimates, we then estimated the engine
the height hk of the car’s trajectory to locally be described by speed r̂k , using the lower bound described in Section III.
a polynomial model. We then used the estimated along track acceleration, along
Also noteworthy is that if measurements of the GPS- track speed, vertical speed, and engine speed to drive the fuel
receiver are uniformly sampled and the measurement noise consumption model described in Section II.
covariance Q = c I, where I denotes the identity matrix Thereafter, we compared the estimated fuel consumption
and c is a positive scalar, then the acceleration estimate of with the (measured) reference fuel consumption ỹ k calculated
the polynomial regression is equivalent to the acceleration from the recorded OBD data. The reference fuel consumption
estimate obtained when filtering the speed signal through the was calculated by dividing the air flow rate data with the air to
minimum noise gain FIR filter differentiator derived in [26]. fuel ratio data, and then low-pass filtering the result through a
Thus, for a computational complexity sensitive application, the moving average filter with a 1 [Hz] bandwidth. The low pass
minimum noise gain FIR filter differentiator may be a good filtering was done to make sure the measured fuel consumption
alternative. However, the benefit of the polynomial regression had the same bandwidth as the motion dynamics estimated via
in (24) is that outliers in the measurements of the GPS-receiver the polynomial regression; the polynomial regression for the
may easily be detected by monitoring the magnitude of the estimation of the along track acceleration and horizontal speed
residual of the fitted measurements. was done using window sizes N = 5 samples (T = 1 [s]) and
If a measurement platform that in addition to a GPS-receiver N = 50 samples (T = 10 [s]), respectively.
also houses inertial sensors or wheel speed sensors is used, In Fig. 6, scatter plots of the estimated versus measured
then it is advisable that instead of the polynomial regression fuel consumption for the three drives are shown. Both the
method, a data fusion scheme such as that in e.g., [27] is estimated fuel consumption when using the true engine speed
used. In general, this will allow the inputs to the the fuel and the estimated engine speed are shown. The root mean
consumption model to be calculated more accurately and at a square error and mean error are also included in the scatter
higher rate. In [28] and [29][pp.435-462], surveys reviewing plots. From the scatter plots it can be seen that on average, thePostprint: to appear in IEEE Trans Instrumentation and Measurement
IEEE TRANSACTIONS , VOL. X, NO. X, XXXX 200X 7
Fuel consumption reference
Mass air flow rate
Mass air flow rate ỹk
Air to fuel ratio LP-filter
OBD Air to fuel ratio
r̃k
Fuel consumption estimator
p p
ṽx,k v̂x,k True RPM +
h̃k Polynomial âpx,k RPM r̂k Fuel Consumption ŷk - estimation error
GPS tk Regression n
v̂z,k Model Model
RPM Lower bound
Prior information
{m, Pbpeak , Vd , φw }
Swedish
Traffic {Vehicle Class, Vd } {A Ca , GF GL }
Register Parameter
Register
Fig. 5: Block diagram of the fuel consumption estimator and the fuel consumption reference system.
proposed instantaneous fuel consumption method works quite Speed versus time
100
well, with a root mean square error on the order of 0.20-0.40
Speed [km/h]
g/s, depending upon the size of the engine and car. Taking 50
the magnitude of the fuel consumption of the car model under
test into account, then in terms of normalized mean square 0
0 2 4 6 8 10 12 14 16
error, i.e., (ỹk − ŷk )2 / (ỹk )2 , the estimation error of the Engine speed versus time
Engine speed [rev/s]
100
proposed method for all three tests is slightly less than 10%. r̃k
r̂k
Further, from the scatter plots it can also be seen that the 50
use of the estimated engine speed instead of the measured
0
engine speed, mainly affects the estimation accuracy at time 0 2 4 6 8 10 12 14 16
Fuel consumption versus time
Fuel consumption [g/s]
instances with low power demands (low fuel consumption).
ỹk
This is because at the time periods with moderate motion ŷk with r̃k
5 ŷk with r̂k
dynamics, the factor that has the largest effect on the power
consumption is the engine speed, see Fig. 3. 0
0 2 4 6 8 10 12 14 16
Time [s]
To illustrate the behavior of the estimator during abrupt ve- Fig. 7: The estimated and measured engine speed and instantaneous fuel
locity changes, the measured and estimated fuel consumption consumption when the Volvo XC70 accelerates from 0 to 100 [km/h].
when the Volvo XC 70 is accelerating from 0 to 100 [km/h]
is shown in Fig. 7. Also shown is the measured and estimated
engine speed. From the figure, we can see that the estimated VI. D ISCUSSION AND CONCLUSIONS
fuel consumptions, both with the estimated and true engine
speeds, agrees quite well with the measured fuel consumption; We have proposed a method to estimate the instantaneous
even though the engine speed estimator, most of the time, fuel consumption of a car only from the data recorded by
underestimates the engine’s speed (especially during the gear a GPS-receiver onboard the car and the vehicle parameters
changes, at around 6, 10, and 13 [s] into the acceleration, accessible through the Swedish vehicle register. The proposed
when the clutch is disengaged and the engine speed peaks estimator is based on a physical model of the power sources
and the car momentarily stops accelerating.). Clearly, during and loads in a car, and is driven by the along-track acceler-
the acceleration, the needed tractive power dominates the total ation, along-track speed, and vertical speed calculated via a
power consumption of the engine, and the power needed to polynomial regression of the data output by the GPS-receiver.
overcome the engine friction only plays a minor role in the We have tested the proposed method on three cars of different
total fuel consumption. makes and sizes, that were driven on a mixture of highway andPostprint: to appear in IEEE Trans Instrumentation and Measurement
IEEE TRANSACTIONS , VOL. X, NO. X, XXXX 200X 8
Estimated fuel consumption: ŷ k [g/s]
3 7 8
Estimated fuel consumption: ŷ k [g/s]
Estimated fuel consumption: ŷ k [g/s]
2.5 RMS error 0.22 [g/s] 6 RMS error 0.28 [g/s] RMS error 0.39 [g/s]
Mean error 0.09 [g/s] Mean error 0.01 [g/s] 6 Mean error 0.14 [g/s]
5
2
4
1.5 4
3
1
2
2
0.5 1
0 0 0
0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 0 2 4 6 8
Measured fuel consumption: ỹ k [g/s] Measured fuel consumption: ỹ k [g/s] Measured fuel consumption: ỹ k [g/s]
(a) Peugeot with true RPM (b) Audi with true RPM (c) Volvo with RPM
3 7 8
Estimated fuel consumption ŷ k [g/s]
Estimated fuel consumption: ŷ k [g/s]
Estimated fuel consumption: ŷ k [g/s]
2.5 6
RMS error 0.22 [g/s] RMS error 0.31 [g/s] RMS error 0.38 [g/s]
Mean error −0.01 [g/s] Mean error −0.12 [g/s] 6 Mean error 0.04 [g/s]
5
2
4
1.5 4
3
1
2
2
0.5 1
0 0 0
0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 0 2 4 6 8
Measured fuel consumption: ỹ k [g/s] Measured fuel consumption: ỹ k [g/s] Measured fuel consumption ỹ k [g/s]
(d) Peugeot with RPM lower bound (e) Audi with RPM lower bound (f) Volvo with RPM lower bound
Fig. 6: Scatter plots of the estimated versus measured fuel instantaneous fuel consumptions, with the true engine speed and the lower bound on the engine
speed, for a Peugeot 206, a Audi A3, and a Volvo XC70.
Indeed, an eco-drive meter based on the proposed fuel
consumption estimation method has, together with other real-
time feedback usable for, so called, usage based insurance
(UBI), been implemented as a part of the virtual vehicle
dashboard illustrated in Fig. 8; refer to [30] for details. UBI is
a type of car insurance where the insurance fee is determined,
not only from traditional factors such as age, gender, vehicle
model, etc., but also from parameters such as distance traveled,
Eco-ness meter time of the trip, location, or driver behavioral parameters such
as smoothness and eco-ness; parameters monitored in real-
time through a plurality of sensors. Accordingly, functionality
such as the eco-drive meter does not only provide the driver
with guidance towards a more eco-friendly driving style with
savings in terms of fuel consumption, but also may imply
a safer driving style with savings on insurance fees as a
byproduct.
Fig. 8: The smartphone used as an extension of the vehicle’s dashboard
for real-time driver feedback. The real-time feedback includes measures of
acceleration, smoothness, braking, cornering, speeding (with respect to the
actual speed limit), and swerving. In addition, a measure of the eco-ness of
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